Here we provide mathematical details on the definition of the DT
invariants \(\Omega _\sigma (\gamma )\) . These invariants are a direct generalisation of the integer BPS invariants of [1]}. From [2]}, the objects of a smooth CY3 dg category \(\mathcal {C}\)  (like the dg category of perfect complexes with compact support on a smooth CY3-fold) form a -1 shifted-symplectic derived stack \(\mathcal {M}\)  in the sense of [3]}. We suppose that \(\mathcal {M}\)  admits an orientation, i.e. a square root of the line bundle \(\,{\rm det}\, (\mathbb {L}_\mathcal {M})\)  given by the determinant of the cotangent complex of \(\mathcal {M}\)  (a canonical orientation was constructed in the case of sheaves with compact support on noncompact CY3-fold in [4]}). For \(\sigma \)  a stability condition on \(\mathcal {C}\) , \(\sigma \) -semistability is a Zariski open condition, hence from [5]} there is an open -1-shifted symplectic substack \(\mathcal {M}_\sigma \hookrightarrow \mathcal {M}\)  of \(\sigma \) -semistable objects. From the definition of semistability, \(\operatorname{Ext}^i(E,E)=0\)  for any \(E\in \mathcal {M}_\sigma \)  and \(i<0\) , hence by [2]} \(\mathbb {T}_{\mathcal {M}_\sigma }|_E=\operatorname{Ext}(E,E)[1]\) , \(\mathcal {M}\)  is a -1 shifted symplectic Artin 1 stack. We then define \(\mathcal {M}_\sigma (\gamma )\)  as the component of \(\mathcal {M}_\sigma \)  of objects of class \(\gamma \)  in the Grothendieck group of \(\mathcal {C}\) . Suppose now that \(\sigma \)  is generic, i.e. that two \(\sigma \) -semistable objects \(E,E^{\prime }\)  of the same phase have collinear charges. For \(\gamma \)  primitive, we define the DT invariants \(\Omega _\sigma (k\gamma ),k\ge 1\)  by:
\({\rm Exp }\left(\sum _{k=1}^\infty \frac{\Omega _\sigma (k\gamma )}{y^{-1}-y}x^k\right)\sum _{k=0}^\infty H_c(\mathcal {M}_\sigma (k\gamma ),P_{\mathcal {M}_\sigma (k\gamma )})x^k\) 
